Non-existence and strong ill-posedness in $C^{k,\beta}$ for the generalized Surface Quasi-geostrophic equation

We consider solutions to the generalized Surface Quasi-geostrophic equation ($\gamma$-SQG) when the velocity is more singular than the active scalar function (i.e. $\gamma\in(0,1)$). In this paper we establish strong ill-posedness in $C^{k,\beta}$ ($k\geq 1$, $\beta\in(0,1]$ and $k+\beta>1+\gamma$) and we also construct solutions in $\mathbb{R}^2$ that initially are in $C^{k,\beta}\cap L^2$ but are not in $C^{k,\beta}$ for $t>0$. Furthermore these solutions stay in $H^{k+\beta+1-2\delta}$ for some small $\delta$ and an arbitrarily long time.